On the classification of simple amenable C∗C*-algebras with finite decomposition rank, II

We prove that every unital stably finite simple amenable $C^*$-algebra $A$ with finite nuclear dimension and with UCT such that every trace is quasi-diagonal has the property that $A\otimes Q$ has generalized tracial rank at most one, where $Q$ is the universal UHF-algebra. Consequently, $A$ is classifiable in the sense of Elliott.Comment: submitted. Some minor upda

On classification of non-unital amenable simple C*-algebras, III

We show that two separable stably projectionless simple C*-algebras A and B with $gTR(A) \le 1$ and $gTR(B) \le 1$ which satisfy the UCT are isomorphic if and only if they have the isomorphic Elliott invariant. A description of Elliott invariant is given. We show all possible simple scaled Elliott invariants can be reached by C*-algebras in the class. We show that these results imply that two separable simple C*-algebras with stable rank one and finite nuclear dimension which satisfy the UCT are isomorphic if and only if they have the same Elliott invariant.Comment: 193 pages, the revision removes the condition of stable rank one in the main theore